Abstract

Correlation dependent, propagation-induced shifts in the generalized spectra of cyclostationary, random fields are predicted. This result generalizes the Wolf shift for stationary fields and is applicable to periodic trains of fast pulses such as might be generated in comb spectroscopy or other mode-locked pulsed systems. Examples illustrate these shifts for intrinsically stationary fields and the fields generated by a mode-locked laser.

© 2009 Optical Society of America

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References

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  1. E. Wolf, Introduction to the Theory of Coherence and Polarization (Cambridge University Press, 2007).
  2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
  3. P. Corkum and Z. Chang, "The Attosecond Revolution," Opt. Photon. News 19, 24-29 (2008).
    [CrossRef]
  4. W. Gardner, A. Napolitano, and L. Paura, "Cyclostationarity: Half a Century of Research," Signal Process. 86, 639-697 (2006).
    [CrossRef]
  5. B. Davis, "Measurable Coherence Theory for Statistically Periodic Fields," Phys. Rev. A 76, 043,843 (2007).
    [CrossRef]
  6. E. Wolf, "Invariance of the Spectrum of Light on Propagation," Phys. Rev. Lett. 56, 1370-1372 (1986).
    [CrossRef] [PubMed]
  7. E. Wolf, "Noncosmological Redshifts of Spectral Lines," Nature (London) 326, 363-365 (1987).
    [CrossRef]
  8. S. Ponomarenko, G. Agrawal, and E. Wolf, "Energy Spectrum of a Nonstationary Ensemble of Pulses," Opt. Lett. 29, 394-396 (2004).
    [CrossRef] [PubMed]
  9. H. Lajunen, P. Vahimaa, and J. Tervo, "Theory of Spatially and Spectrally Partially Coherent Pulses," J. Opt. Soc. Am. A 22, 1536-1545 (2005).
    [CrossRef]
  10. R. Gase and M. Schubert, "On the Determination of Spectral Properties of Non-stationary Radiation," J. Mod. Opt. 29, 1331-1347 (1982).
  11. R. Schoonover, B. Davis, R. Bartels, and P. Carney, "Optical Interferometry with Pulsed Fields," J. Mod. Opt. 55, 1541-1556 (2008).
    [CrossRef]
  12. A. Kubo, K. Onda, H. Petek, Z. Sun, Y. Jung, and H. Kim, "Femtosecond Imaging of Surface Plasmon Dynamics in a Nanostructured Silver Film," Nano Lett 5, 1123-1127 (2005).
    [CrossRef] [PubMed]
  13. Y. Liau, A. Unterreiner, Q. Chang, and N. Scherer, "Ultrafast Dephasing of Single Nanoparticles Studied by Two-pulse Second-order Interferometry," J. Phys. Chem. B 105, 2135-2142 (2001).
    [CrossRef]
  14. T. Stievater, X. Li, D. Steel, D. Gammon, D. Katzer, D. Park, C. Piermarocchi, and L. Sham, "Rabi Oscillations of Excitons in Single Quantum Dots," Phys. Rev. Lett. 87, 133,603 (2001).
    [CrossRef]
  15. J. Reichert, R. Holzwarth, T. Udem, and T. H¨ansch,"Measuring the Frequency of Light with Mode-locked Lasers," Opt. Commun. 172, 59-68 (1999).
    [CrossRef]
  16. D. Jones, S. Diddams, J. Ranka, A. Stentz, R. Windeler, J. Hall, and S. Cundiff, "Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis," Science (London) 288, 635 (2000).
  17. B. Saleh and M. Teich, Fundamentals of Photonics (New York: Wiley-Interscience, 1991).
    [CrossRef]
  18. S. Jacobs, "The Optical Heterodyne," Electronics 36, 29 (1963).
  19. A. Siegman, S. Harris, and B. McMurtry, "Optical Heterodyning and Optical Demodulation at Microwave Frequencies," Optical Masers pp. 511-527 (1963).
  20. T. Young, "The Bakerian Lecture. Experiments and Calculations Relative to Physical Optics," Phil. Trans. R. Soc. Lond 94 (1804).

2008 (2)

R. Schoonover, B. Davis, R. Bartels, and P. Carney, "Optical Interferometry with Pulsed Fields," J. Mod. Opt. 55, 1541-1556 (2008).
[CrossRef]

P. Corkum and Z. Chang, "The Attosecond Revolution," Opt. Photon. News 19, 24-29 (2008).
[CrossRef]

2007 (1)

B. Davis, "Measurable Coherence Theory for Statistically Periodic Fields," Phys. Rev. A 76, 043,843 (2007).
[CrossRef]

2006 (1)

W. Gardner, A. Napolitano, and L. Paura, "Cyclostationarity: Half a Century of Research," Signal Process. 86, 639-697 (2006).
[CrossRef]

2005 (2)

A. Kubo, K. Onda, H. Petek, Z. Sun, Y. Jung, and H. Kim, "Femtosecond Imaging of Surface Plasmon Dynamics in a Nanostructured Silver Film," Nano Lett 5, 1123-1127 (2005).
[CrossRef] [PubMed]

H. Lajunen, P. Vahimaa, and J. Tervo, "Theory of Spatially and Spectrally Partially Coherent Pulses," J. Opt. Soc. Am. A 22, 1536-1545 (2005).
[CrossRef]

2004 (1)

2001 (2)

Y. Liau, A. Unterreiner, Q. Chang, and N. Scherer, "Ultrafast Dephasing of Single Nanoparticles Studied by Two-pulse Second-order Interferometry," J. Phys. Chem. B 105, 2135-2142 (2001).
[CrossRef]

T. Stievater, X. Li, D. Steel, D. Gammon, D. Katzer, D. Park, C. Piermarocchi, and L. Sham, "Rabi Oscillations of Excitons in Single Quantum Dots," Phys. Rev. Lett. 87, 133,603 (2001).
[CrossRef]

2000 (1)

D. Jones, S. Diddams, J. Ranka, A. Stentz, R. Windeler, J. Hall, and S. Cundiff, "Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis," Science (London) 288, 635 (2000).

1999 (1)

J. Reichert, R. Holzwarth, T. Udem, and T. H¨ansch,"Measuring the Frequency of Light with Mode-locked Lasers," Opt. Commun. 172, 59-68 (1999).
[CrossRef]

1987 (1)

E. Wolf, "Noncosmological Redshifts of Spectral Lines," Nature (London) 326, 363-365 (1987).
[CrossRef]

1986 (1)

E. Wolf, "Invariance of the Spectrum of Light on Propagation," Phys. Rev. Lett. 56, 1370-1372 (1986).
[CrossRef] [PubMed]

1982 (1)

R. Gase and M. Schubert, "On the Determination of Spectral Properties of Non-stationary Radiation," J. Mod. Opt. 29, 1331-1347 (1982).

1963 (1)

S. Jacobs, "The Optical Heterodyne," Electronics 36, 29 (1963).

1804 (1)

T. Young, "The Bakerian Lecture. Experiments and Calculations Relative to Physical Optics," Phil. Trans. R. Soc. Lond 94 (1804).

Agrawal, G.

Bartels, R.

R. Schoonover, B. Davis, R. Bartels, and P. Carney, "Optical Interferometry with Pulsed Fields," J. Mod. Opt. 55, 1541-1556 (2008).
[CrossRef]

Carney, P.

R. Schoonover, B. Davis, R. Bartels, and P. Carney, "Optical Interferometry with Pulsed Fields," J. Mod. Opt. 55, 1541-1556 (2008).
[CrossRef]

Chang, Q.

Y. Liau, A. Unterreiner, Q. Chang, and N. Scherer, "Ultrafast Dephasing of Single Nanoparticles Studied by Two-pulse Second-order Interferometry," J. Phys. Chem. B 105, 2135-2142 (2001).
[CrossRef]

Chang, Z.

P. Corkum and Z. Chang, "The Attosecond Revolution," Opt. Photon. News 19, 24-29 (2008).
[CrossRef]

Corkum, P.

P. Corkum and Z. Chang, "The Attosecond Revolution," Opt. Photon. News 19, 24-29 (2008).
[CrossRef]

Cundiff, S.

D. Jones, S. Diddams, J. Ranka, A. Stentz, R. Windeler, J. Hall, and S. Cundiff, "Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis," Science (London) 288, 635 (2000).

Davis, B.

R. Schoonover, B. Davis, R. Bartels, and P. Carney, "Optical Interferometry with Pulsed Fields," J. Mod. Opt. 55, 1541-1556 (2008).
[CrossRef]

B. Davis, "Measurable Coherence Theory for Statistically Periodic Fields," Phys. Rev. A 76, 043,843 (2007).
[CrossRef]

Diddams, S.

D. Jones, S. Diddams, J. Ranka, A. Stentz, R. Windeler, J. Hall, and S. Cundiff, "Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis," Science (London) 288, 635 (2000).

Gammon, D.

T. Stievater, X. Li, D. Steel, D. Gammon, D. Katzer, D. Park, C. Piermarocchi, and L. Sham, "Rabi Oscillations of Excitons in Single Quantum Dots," Phys. Rev. Lett. 87, 133,603 (2001).
[CrossRef]

Gardner, W.

W. Gardner, A. Napolitano, and L. Paura, "Cyclostationarity: Half a Century of Research," Signal Process. 86, 639-697 (2006).
[CrossRef]

Gase, R.

R. Gase and M. Schubert, "On the Determination of Spectral Properties of Non-stationary Radiation," J. Mod. Opt. 29, 1331-1347 (1982).

H¨ansch, T.

J. Reichert, R. Holzwarth, T. Udem, and T. H¨ansch,"Measuring the Frequency of Light with Mode-locked Lasers," Opt. Commun. 172, 59-68 (1999).
[CrossRef]

Hall, J.

D. Jones, S. Diddams, J. Ranka, A. Stentz, R. Windeler, J. Hall, and S. Cundiff, "Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis," Science (London) 288, 635 (2000).

Holzwarth, R.

J. Reichert, R. Holzwarth, T. Udem, and T. H¨ansch,"Measuring the Frequency of Light with Mode-locked Lasers," Opt. Commun. 172, 59-68 (1999).
[CrossRef]

Jacobs, S.

S. Jacobs, "The Optical Heterodyne," Electronics 36, 29 (1963).

Jones, D.

D. Jones, S. Diddams, J. Ranka, A. Stentz, R. Windeler, J. Hall, and S. Cundiff, "Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis," Science (London) 288, 635 (2000).

Jung, Y.

A. Kubo, K. Onda, H. Petek, Z. Sun, Y. Jung, and H. Kim, "Femtosecond Imaging of Surface Plasmon Dynamics in a Nanostructured Silver Film," Nano Lett 5, 1123-1127 (2005).
[CrossRef] [PubMed]

Katzer, D.

T. Stievater, X. Li, D. Steel, D. Gammon, D. Katzer, D. Park, C. Piermarocchi, and L. Sham, "Rabi Oscillations of Excitons in Single Quantum Dots," Phys. Rev. Lett. 87, 133,603 (2001).
[CrossRef]

Kim, H.

A. Kubo, K. Onda, H. Petek, Z. Sun, Y. Jung, and H. Kim, "Femtosecond Imaging of Surface Plasmon Dynamics in a Nanostructured Silver Film," Nano Lett 5, 1123-1127 (2005).
[CrossRef] [PubMed]

Kubo, A.

A. Kubo, K. Onda, H. Petek, Z. Sun, Y. Jung, and H. Kim, "Femtosecond Imaging of Surface Plasmon Dynamics in a Nanostructured Silver Film," Nano Lett 5, 1123-1127 (2005).
[CrossRef] [PubMed]

Lajunen, H.

Li, X.

T. Stievater, X. Li, D. Steel, D. Gammon, D. Katzer, D. Park, C. Piermarocchi, and L. Sham, "Rabi Oscillations of Excitons in Single Quantum Dots," Phys. Rev. Lett. 87, 133,603 (2001).
[CrossRef]

Liau, Y.

Y. Liau, A. Unterreiner, Q. Chang, and N. Scherer, "Ultrafast Dephasing of Single Nanoparticles Studied by Two-pulse Second-order Interferometry," J. Phys. Chem. B 105, 2135-2142 (2001).
[CrossRef]

Napolitano, A.

W. Gardner, A. Napolitano, and L. Paura, "Cyclostationarity: Half a Century of Research," Signal Process. 86, 639-697 (2006).
[CrossRef]

Onda, K.

A. Kubo, K. Onda, H. Petek, Z. Sun, Y. Jung, and H. Kim, "Femtosecond Imaging of Surface Plasmon Dynamics in a Nanostructured Silver Film," Nano Lett 5, 1123-1127 (2005).
[CrossRef] [PubMed]

Park, D.

T. Stievater, X. Li, D. Steel, D. Gammon, D. Katzer, D. Park, C. Piermarocchi, and L. Sham, "Rabi Oscillations of Excitons in Single Quantum Dots," Phys. Rev. Lett. 87, 133,603 (2001).
[CrossRef]

Paura, L.

W. Gardner, A. Napolitano, and L. Paura, "Cyclostationarity: Half a Century of Research," Signal Process. 86, 639-697 (2006).
[CrossRef]

Petek, H.

A. Kubo, K. Onda, H. Petek, Z. Sun, Y. Jung, and H. Kim, "Femtosecond Imaging of Surface Plasmon Dynamics in a Nanostructured Silver Film," Nano Lett 5, 1123-1127 (2005).
[CrossRef] [PubMed]

Piermarocchi, C.

T. Stievater, X. Li, D. Steel, D. Gammon, D. Katzer, D. Park, C. Piermarocchi, and L. Sham, "Rabi Oscillations of Excitons in Single Quantum Dots," Phys. Rev. Lett. 87, 133,603 (2001).
[CrossRef]

Ponomarenko, S.

Ranka, J.

D. Jones, S. Diddams, J. Ranka, A. Stentz, R. Windeler, J. Hall, and S. Cundiff, "Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis," Science (London) 288, 635 (2000).

Reichert, J.

J. Reichert, R. Holzwarth, T. Udem, and T. H¨ansch,"Measuring the Frequency of Light with Mode-locked Lasers," Opt. Commun. 172, 59-68 (1999).
[CrossRef]

Scherer, N.

Y. Liau, A. Unterreiner, Q. Chang, and N. Scherer, "Ultrafast Dephasing of Single Nanoparticles Studied by Two-pulse Second-order Interferometry," J. Phys. Chem. B 105, 2135-2142 (2001).
[CrossRef]

Schoonover, R.

R. Schoonover, B. Davis, R. Bartels, and P. Carney, "Optical Interferometry with Pulsed Fields," J. Mod. Opt. 55, 1541-1556 (2008).
[CrossRef]

Schubert, M.

R. Gase and M. Schubert, "On the Determination of Spectral Properties of Non-stationary Radiation," J. Mod. Opt. 29, 1331-1347 (1982).

Sham, L.

T. Stievater, X. Li, D. Steel, D. Gammon, D. Katzer, D. Park, C. Piermarocchi, and L. Sham, "Rabi Oscillations of Excitons in Single Quantum Dots," Phys. Rev. Lett. 87, 133,603 (2001).
[CrossRef]

Steel, D.

T. Stievater, X. Li, D. Steel, D. Gammon, D. Katzer, D. Park, C. Piermarocchi, and L. Sham, "Rabi Oscillations of Excitons in Single Quantum Dots," Phys. Rev. Lett. 87, 133,603 (2001).
[CrossRef]

Stentz, A.

D. Jones, S. Diddams, J. Ranka, A. Stentz, R. Windeler, J. Hall, and S. Cundiff, "Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis," Science (London) 288, 635 (2000).

Stievater, T.

T. Stievater, X. Li, D. Steel, D. Gammon, D. Katzer, D. Park, C. Piermarocchi, and L. Sham, "Rabi Oscillations of Excitons in Single Quantum Dots," Phys. Rev. Lett. 87, 133,603 (2001).
[CrossRef]

Sun, Z.

A. Kubo, K. Onda, H. Petek, Z. Sun, Y. Jung, and H. Kim, "Femtosecond Imaging of Surface Plasmon Dynamics in a Nanostructured Silver Film," Nano Lett 5, 1123-1127 (2005).
[CrossRef] [PubMed]

Tervo, J.

Udem, T.

J. Reichert, R. Holzwarth, T. Udem, and T. H¨ansch,"Measuring the Frequency of Light with Mode-locked Lasers," Opt. Commun. 172, 59-68 (1999).
[CrossRef]

Unterreiner, A.

Y. Liau, A. Unterreiner, Q. Chang, and N. Scherer, "Ultrafast Dephasing of Single Nanoparticles Studied by Two-pulse Second-order Interferometry," J. Phys. Chem. B 105, 2135-2142 (2001).
[CrossRef]

Vahimaa, P.

Windeler, R.

D. Jones, S. Diddams, J. Ranka, A. Stentz, R. Windeler, J. Hall, and S. Cundiff, "Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis," Science (London) 288, 635 (2000).

Wolf, E.

S. Ponomarenko, G. Agrawal, and E. Wolf, "Energy Spectrum of a Nonstationary Ensemble of Pulses," Opt. Lett. 29, 394-396 (2004).
[CrossRef] [PubMed]

E. Wolf, "Noncosmological Redshifts of Spectral Lines," Nature (London) 326, 363-365 (1987).
[CrossRef]

E. Wolf, "Invariance of the Spectrum of Light on Propagation," Phys. Rev. Lett. 56, 1370-1372 (1986).
[CrossRef] [PubMed]

Young, T.

T. Young, "The Bakerian Lecture. Experiments and Calculations Relative to Physical Optics," Phil. Trans. R. Soc. Lond 94 (1804).

Electronics (1)

S. Jacobs, "The Optical Heterodyne," Electronics 36, 29 (1963).

J. Mod. Opt. (2)

R. Gase and M. Schubert, "On the Determination of Spectral Properties of Non-stationary Radiation," J. Mod. Opt. 29, 1331-1347 (1982).

R. Schoonover, B. Davis, R. Bartels, and P. Carney, "Optical Interferometry with Pulsed Fields," J. Mod. Opt. 55, 1541-1556 (2008).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Phys. Chem. B (1)

Y. Liau, A. Unterreiner, Q. Chang, and N. Scherer, "Ultrafast Dephasing of Single Nanoparticles Studied by Two-pulse Second-order Interferometry," J. Phys. Chem. B 105, 2135-2142 (2001).
[CrossRef]

Nano Lett (1)

A. Kubo, K. Onda, H. Petek, Z. Sun, Y. Jung, and H. Kim, "Femtosecond Imaging of Surface Plasmon Dynamics in a Nanostructured Silver Film," Nano Lett 5, 1123-1127 (2005).
[CrossRef] [PubMed]

Nature (London) (1)

E. Wolf, "Noncosmological Redshifts of Spectral Lines," Nature (London) 326, 363-365 (1987).
[CrossRef]

Opt. Commun. (1)

J. Reichert, R. Holzwarth, T. Udem, and T. H¨ansch,"Measuring the Frequency of Light with Mode-locked Lasers," Opt. Commun. 172, 59-68 (1999).
[CrossRef]

Opt. Lett. (1)

Opt. Photon. News (1)

P. Corkum and Z. Chang, "The Attosecond Revolution," Opt. Photon. News 19, 24-29 (2008).
[CrossRef]

Phil. Trans. R. Soc. Lond (1)

T. Young, "The Bakerian Lecture. Experiments and Calculations Relative to Physical Optics," Phil. Trans. R. Soc. Lond 94 (1804).

Phys. Rev. A (1)

B. Davis, "Measurable Coherence Theory for Statistically Periodic Fields," Phys. Rev. A 76, 043,843 (2007).
[CrossRef]

Phys. Rev. Lett. (2)

E. Wolf, "Invariance of the Spectrum of Light on Propagation," Phys. Rev. Lett. 56, 1370-1372 (1986).
[CrossRef] [PubMed]

T. Stievater, X. Li, D. Steel, D. Gammon, D. Katzer, D. Park, C. Piermarocchi, and L. Sham, "Rabi Oscillations of Excitons in Single Quantum Dots," Phys. Rev. Lett. 87, 133,603 (2001).
[CrossRef]

Science (London) (1)

D. Jones, S. Diddams, J. Ranka, A. Stentz, R. Windeler, J. Hall, and S. Cundiff, "Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis," Science (London) 288, 635 (2000).

Signal Process. (1)

W. Gardner, A. Napolitano, and L. Paura, "Cyclostationarity: Half a Century of Research," Signal Process. 86, 639-697 (2006).
[CrossRef]

Other (4)

E. Wolf, Introduction to the Theory of Coherence and Polarization (Cambridge University Press, 2007).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

B. Saleh and M. Teich, Fundamentals of Photonics (New York: Wiley-Interscience, 1991).
[CrossRef]

A. Siegman, S. Harris, and B. McMurtry, "Optical Heterodyning and Optical Demodulation at Microwave Frequencies," Optical Masers pp. 511-527 (1963).

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Figures (1)

Fig. 1.
Fig. 1.

(Color online) The normalized spectral density C̃0(ω) and the spectral correlation function |C̃4(ω|) normalized by its peak, at the source (dashed lines in red) and at a point in the far zone P = (100,0,12000) mm (shown in black) for the case when there are three sources, located at (−100,0,0) mm, (0,0,0) mm, and (100,0,0) mm. The peak of C̃0(ω) is at a higher frequency than the peak of |C̃4(ω)|. The top two panels contain plots of C̃0 and |C̃4| for a three-source system with μ = 0.8, the repetition frequency is taken to be 5×1013 rad/s, the bandwidth of the coherence and the bandwidth of the spectrum are both 20% of ̃ the center frequency ωc = 5×1015 rad/s. The middle two panels contain plots of C̃0 and |C̃4| for the same three-source system as above, only with μ= 0.1. The bottom two panels contain plots of C̃0 and |C̃4| for a three-source system that differs from the one in the middle panels by changing the repetition frequency from 5 ×1013 rad/s to 2.5×1014 rad/s.

Equations (30)

Equations on this page are rendered with MathJax. Learn more.

Γ σ ( r 1 , r 2 , t τ , t ) = σ * ( r 1 , t τ ) σ ( r 2 , t ) ,
Γ σ ( r 1 , r 2 , t τ , t ) = n = C n ( r 1 , r 2 , τ ) e i ω 0 nt ,
W σ ( r 1 , r 2 , ω 1 , ω 2 ) = d t 1 d t 2 Γ σ ( r 1 , r 2 , t 1 , t 2 ) e i ( ω 2 t 2 ω 1 t 1 )
W σ ( r 1 , r 2 , ω , ω + Ω ) = n C ˜ n ( r 1 , r 2 , ω ) δ ( Ω n ω 0 ) ,
( 1 2 + k 1 2 ) ( 2 2 + k 2 2 ) W R ( r 1 , r 2 , ω , ω + Ω ) = ( 4 π ) 2 W σ ( r 1 , r 2 , ω , ω + Ω ) ,
W ( ) ( r 1 , r 2 , ω , ω + Ω ) = e i ( k 2 r 2 k 1 r 1 ) r 1 r 2 W σ ( k 1 r ̂ 1 , k 2 r ̂ 2 , ω , ω + Ω ) ,
W ( ) ( r 1 , r 2 , ω , ω + Ω ) = n e i ( k 2 , n r 2 k 1 r 1 ) r 1 r 2 C ˜ n ( k 1 r ̂ 1 , k 2 , n r ̂ 2 , ω )
× δ ( Ω n ω 0 )
S ( ) ( r , ω ) = C ˜ 0 ( k r ̂ , k r ̂ , ω ) r 2 .
W ¯ ( r 1 , r 2 , ω ) = Γ ¯ ( r 1 , r 2 , τ ) e iωτ .
S ( ) ( r , ω ) = 1 r 2 n h n 2 W ¯ ( k r ̂ , k r ̂ , ω n ω 0 ) ,
S ( ) ( r , ω ) = D ( k r ̂ , k r ̂ ) r 2 n h n 2 A ( ω n ω 0 ) .
σ ( r , t ) = n = p = 1 M σ p ( t ) h n exp ( in ω 0 t ) δ ( 3 ) ( r r ( p ) )
σ ˜ p * ( ω ) σ ˜ q ( ω′ ) = S ( ω ) μ pq ( ω ) δ ( ω ω′ ) ,
W ( ) ( r 1 , r 2 , ω , ω + Ω ) = m , n p , q = 1 M h n * h m + n e i k 1 r 1 r ( p ) r 1 r ( p ) e i k 2 , m r 2 r ( q ) r 2 r ( q )
× S ( ω n ω 0 ) μ pq ( ω n ω 0 ) δ ( Ω n ω 0 ) .
μ ij ( ω ) = { 1 i = j μ exp ( ( ω ω c ) 2 2 χ 2 ) i j ,
C ˜ m ( r 1 , r 2 , ω ) = F ( r 1 , r 2 ) n α nn + m δ ( ω ω c + n ω 0 )
I ( ρ , t ; τ ) C 0 ( ρ , ρ , 0 ) + C 0 ( ρ , ρ , τ ) cos α 0 ( ρ , τ )
+ 2 m = 1 { C m ( ρ , ρ , 0 ) cos ( m ω 0 τ 2 ) cos [ m ω 0 ( t τ / 2 ) α m ( ρ , 0 ) ]
+ C m ( ρ , ρ , τ ) cos [ m ω 0 t α m ( ρ , τ ) ] + C m ( ρ , ρ , τ ) cos [ m ω 0 t + α m ( ρ , τ ) ] } ,
I 12 ( t ; τ ) C 0 ( 11 ) ( 0 ) + C 0 ( 22 ) ( 0 ) + 2 C 0 ( 12 ) ( τ ) cos [ α 0 ( 12 ) ( τ ) ]
+ 2 m = 1 { C m ( 11 ) ( 0 ) cos [ m ω 0 ( t R 1 / c ) + α m ( 11 ) ( 0 ) ]
+ C m ( 22 ) ( 0 ) cos [ m ω 0 ( t R 2 / c ) + α m ( 22 ) ( 0 ) ]
+ C m ( 12 ) ( τ ) cos [ m ω 0 ( t R 2 / c ) + α m ( 12 ) ( τ ) ]
+ C m ( 21 ) ( τ ) cos [ m ω 0 ( t R 1 / c ) + α m ( 21 ) ( τ ) ] } ,
r 2 ( ρ , t ; τ ) F 0 ( r ̂ , 0 ) + F 0 ( r ̂ , 0 ) cos ϕ 0 ( r ̂ , τ )
+ 2 m = 1 { F m ( r ̂ , 0 ) cos ( m ω 0 τ 2 ) cos [ m ω 0 ( t τ / 2 r / c ) ϕ m ( r ̂ , 0 ) ]
+ F m ( r ̂ , τ ) cos [ m ω 0 ( t r / c ) ϕ m ( r ̂ , τ ) ]
+ F m ( r ̂ , τ ) cos [ m ω 0 ( t r / c ) ϕ m ( r ̂ , τ ) ] }

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